Finite Group Databasebeta

← C59×#12700 #28424809

Subgroups — C59×#12700 #28424809

Order 15104.

1018 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊Q8×C59C211yesmaximal, characteristic
2C2×C59×#356C214yesmaximal, complemented
3C59×#1784C212yesmaximal
4C23×Q8×C59C411yescharacteristic
5C23×Q8×C59C2211yescharacteristic
6C2×C4⋊Q8×C59—22no—
7C2×C4⋊Q8×C59—28no—
8C2×C4⋊Q8×C59C412yes—
9C2×C4⋊Q8×C59C2212yes—
10C22×C4⋊C4×C59—22no—
11C22×C42×C59C411yescharacteristic
12C2×#101×C59—22no—
13C2×#101×C59C2214yes—
14C59×#407—22no—
15C59×#356—28no—
16C22×Q8×C59—24no—
17C22×Q8×C59—24no—
18C22×Q8×C59—21no—
19C22×Q8×C59—42no—
20C22×Q8×C59—42no—
21C22×Q8×C59—42no—
22C22×Q8×C59—42no—
23C22×Q8×C59C2×C412yes—
24C22×Q8×C59C2×C411yescharacteristic
25C22×Q8×C59D412yes—
26C22×Q8×C59C2311yescharacteristic
27C23×C4×C59—21no—
28C23×C4×C59—21no—
29C23×C4×C59C2×C411yescharacteristic
30C23×C4×C59D412yes—
31C2×M16×C59—24no—
32C4⋊Q8×C59—216no—
33C4⋊Q8×C59—24no—
34C4⋊Q8×C59—24no—
35C4⋊Q8×C59—24no—
36C4⋊Q8×C59D418yes—
37C2×C4⋊C4×C59—216no—
38C2×C4⋊C4×C59—24no—
39C2×C4⋊C4×C59—24no—
40C2×C42×C59—21no—
41C2×C42×C59—24no—
42C2×C42×C59C2×C412yes—
43#101×C59—28no—
44#101×C59—44no—
45C22⋊C8×C59—44no—
46C24×C59C22⋊C411yescharacteristic
47C2×Q8×C59—24no—
48C2×Q8×C59—24no—
49C2×Q8×C59—22no—
50C2×Q8×C59—22no—
51C2×Q8×C59—24no—
52C2×Q8×C59—41no—
53C2×Q8×C59—41no—
54C2×Q8×C59—42no—
55C2×Q8×C59—41no—
56C2×Q8×C59—41no—
57C2×Q8×C59—416no—
58C2×Q8×C59—416no—
59C2×Q8×C59—42no—
60C2×Q8×C59—42no—
61C2×Q8×C59—42no—
62C2×Q8×C59—42no—
63C2×Q8×C59C22⋊C414yes—
64C2×Q8×C59C2×D414yes—
65C22×C4×C59—24no—
66C22×C4×C59—28no—
67C22×C4×C59—22no—
68C22×C4×C59—22no—
69C22×C4×C59—22no—
70C22×C4×C59—22no—
71C22×C4×C59—24no—
72C22×C4×C59—24no—
73C22×C4×C59—22no—
74C22×C4×C59—24no—
75C22×C4×C59—42no—
76C22×C4×C59—42no—
77C22×C4×C59—42no—
78C22×C4×C59C22⋊C412yes—
79C22×C4×C59C22⋊C411yescharacteristic
80C22×C4×C59C22×C411yescharacteristic
81C22×C4×C59C2×D412yes—
82M16×C59—44no—
83M16×C59—44no—
84C2×C8×C59—44no—
85C4⋊C4×C59—216no—
86C4⋊C4×C59—216no—
87C42×C59—24no—
88C42×C59—22no—
89C42×C59C22⋊C414yes—
90C23×C59—24no—
91C23×C59—21no—
92C23×C59—21no—
93C23×C59#9912yes—
94C23×C59C2×C22⋊C411yescharacteristic
95Q8×C59—44no—
96Q8×C59—48no—
97Q8×C59—44no—
98Q8×C59—48no—
99Q8×C59—44no—
100Q8×C59—44no—
101C2×C4×C59—216no—
102C2×C4×C59—28no—
103C2×C4×C59—28no—
104C2×C4×C59—21no—
105C2×C4×C59—24no—
106C2×C4×C59—22no—
107C2×C4×C59—21no—
108C2×C4×C59—22no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—24no—
112C2×C4×C59—22no—
113C2×C4×C59—22no—
114C2×C4×C59—22no—
115C2×C4×C59—48no—
116C2×C4×C59—48no—
117C2×C4×C59—42no—
118C2×C4×C59—42no—
119C2×C4×C59#9912yes—
120C2×C4×C59C2×C22⋊C412yes—
121C2×C4×C59#12014yes—
122C8×C59—84no—
123#12700C5911yesmaximal, characteristic, direct product factor, Sylow
124C22×C59—24no—
125C22×C59—24no—
126C22×C59—22no—
127C22×C59—22no—
128C22×C59—22no—
129C22×C59#35614yescomplemented
130C22×C59#37911yescharacteristic
131C22×C59C2×#9911yescharacteristic
132C22×C59C2×#9911yescharacteristic
133C4×C59—28no—
134C4×C59—24no—
135C4×C59—22no—
136C4×C59—22no—
137C4×C59—42no—
138C4×C59—44no—
139C22×C4⋊Q8C2×C5911yescharacteristic
140C2×#356C2×C5914yescomplemented
141#1784C2×C5912yes—
142C2×C59—21no—
143C2×C59—24no—
144C2×C59—21no—
145C2×C59#178111yescharacteristic
146C2×C59C2×#35612yes—
147C23×Q8C4×C5911yescharacteristic
148C23×Q8C22×C5911yescharacteristic
149C2×C4⋊Q8—22no—
150C2×C4⋊Q8—28no—
151C2×C4⋊Q8C4×C5912yes—
152C2×C4⋊Q8C22×C5912yes—
153C22×C4⋊C4—22no—
154C22×C42C4×C5911yescharacteristic
155C2×#101—22no—
156C2×#101C22×C5914yes—
157#407—22no—
158#356—28no—
159C59#1270011yescharacteristic, direct product factor, Sylow
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—21no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8C2×C4×C5912yes—
168C22×Q8C2×C4×C5911yescharacteristic
169C22×Q8D4×C5912yes—
170C22×Q8C23×C5911yescharacteristic
171C23×C4—21no—
172C23×C4—21no—
173C23×C4C2×C4×C5911yescharacteristic
174C23×C4D4×C5912yes—
175C2×M16—24no—
176C4⋊Q8—216no—
177C4⋊Q8—24no—
178C4⋊Q8—24no—
179C4⋊Q8—24no—
180C4⋊Q8D4×C5918yes—
181C2×C4⋊C4—216no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C42—21no—
185C2×C42—24no—
186C2×C42C2×C4×C5912yes—
187#101—28no—
188#101—44no—
189C22⋊C8—44no—
190C24C22⋊C4×C5911yescharacteristic
191C2×Q8—24no—
192C2×Q8—24no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—24no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—42no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—416no—
202C2×Q8—416no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8C22⋊C4×C5914yes—
208C2×Q8C2×D4×C5914yes—
209C22×C4—24no—
210C22×C4—28no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4C22⋊C4×C5912yes—
223C22×C4C22⋊C4×C5911yescharacteristic
224C22×C4C22×C4×C5911yescharacteristic
225C22×C4C2×D4×C5912yes—
226M16—44no—
227M16—44no—
228C2×C8—44no—
229C4⋊C4—216no—
230C4⋊C4—216no—
231C42—24no—
232C42—22no—
233C42C22⋊C4×C5914yes—
234C23—24no—
235C23—21no—
236C23—21no—
237C23#99×C5912yes—
238C23C2×C22⋊C4×C5911yescharacteristic
239Q8—44no—
240Q8—48no—
241Q8—44no—
242Q8—48no—
243Q8—44no—
244Q8—44no—
245C2×C4—216no—
246C2×C4—28no—
247C2×C4—28no—
248C2×C4—21no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—48no—
260C2×C4—48no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4#99×C5912yes—
264C2×C4C2×C22⋊C4×C5912yes—
265C2×C4#120×C5914yes—
266C8—84no—
267C22—24no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22C59×#35614yescomplemented
273C22C59×#37911yescharacteristic
274C22C2×#99×C5911yescharacteristic
275C22C2×#99×C5911yescharacteristic
276C4—28no—
277C4—24no—
278C4—22no—
279C4—22no—
280C4—42no—
281C4—44no—
282C2—21no—
283C2—24no—
284C2—21no—
285C2C59×#178111yescharacteristic
286C2C2×C59×#35612yes—
subgroupG/H
center, socleC22×C59C2×#99
commutatorC22×C4C22×C4×C59
Frattini, cosocleC22×Q8C23×C59
prime radicalC24×C59C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C24×C59 ◁ C23×Q8×C59 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C59×#356C2semidirect
123. #12700C59direct (N × H)
129. C22×C59#356semidirect
140. C2×#356C2×C59semidirect
159. C59#12700direct (N × H)
272. C22C59×#356semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C4⋊Q8×C592
C2×C59×#3562
C59×#17842
#1270059